How to estimate the total if only a fraction of the data points are known?

Question

I understand that this is more of a statistics question, but there is no "Statistics" category.

Say some kind of machine is produced and numbered (serial number) and I encounter a couple of machines in the "field". How can I estimate the total production from the numbers encountered?

Is there literature on this?

Answer 1

Neat question!!!

Stats is math as far as I am concerned, so this is the right place =)

Let's suppose you have k sample serial numbers, order them from smallest to largest, and call them s_1, s_2, ..., s_k, where s_1 < s_2 < ... < s_k. Then a good estimate of the total production is s_k + (s_k - s_(k-1)). That is, it can be shown that E[s_k + (s_k - s_(k-1))] = the total production of machines.

An even better estimate (that is one with the same expected value, but a lower variance) for the total production is s_k + (s_k - s_1)/(k-1).

Alas I do not know a good reference offhand. I know there were a couple papers written on this in the 1950's about how this method was used by British Intelligence during WWII to estimate German military production, but unfortunately I do not recall the names of the authors, or the journal.